Seifert–Weber space

Example

Take a solid regular dodecahedron. Glue each face to the opposite face after a turn of $3/10$ of a full turn, that is, by the angle $3\pi/5$. The result is the Seifert–Weber space $W$, a closed orientable 3-manifold.

Under these gluings, the $30$ edges of the dodecahedron fall into $6$ classes of $5$ edges, and all $20$ vertices become one point. The cell structure has $1$ vertex, $6$ edges, $6$ faces and $1$ cell. So its Euler characteristic is $1 - 6 + 6 - 1 = 0$, as it must be for a closed $3$-manifold. In hyperbolic space $\mathbb{H}^3$ there is a regular dodecahedron with dihedral angles $2\pi/5$. With this dodecahedron the $5$ angles around each edge class add up to $2\pi$, and the gluing maps are isometries. So $W$ is a closed hyperbolic 3-manifold.